Context. Quiet-Sun Ellerman bombs (QSEBs) are small-scale  magnetic reconnection events in the lower atmosphere of the quiet Sun.  Recent work has shown that a small percentage of them can occur  co-spatially and co-temporally with ultraviolet (UV) brightenings in the  transition region.
Aims. We aim to understand how the magnetic topologies  associated with closely occurring QSEBs and UV brightenings can  facilitate energy transport and connect these events.
Methods. We used high-resolution Hβ observations from the Swedish 1-m Solar Telescope (SST) and detected QSEBs using k-means  clustering. We obtained the magnetic field topology from potential  field extrapolations using spectro-polarimetric data in the photospheric  Fe I 6173 Å line. To detect UV  brightenings, we used coordinated and co-aligned data from the Interface  Region Imaging Spectrograph (IRIS) and imposed a threshold of 5σ above the median background on the (IRIS) 1400 Å slit-jaw image channel.
Results. We identify four distinct magnetic configurations  that associate QSEBs with UV brightenings, including a simple dipole  configuration and more complex fan-spine topologies with a 3D magnetic  null point. In the fan-spine topology, the UV brightenings occur near  the 3D null point, while QSEBs can be found close to the footpoints of  the outer spine, the inner spine, and the fan surface. The height of the  3D null varies between 0.2 Mm and 2.6 Mm, depending on the magnetic  field strength in the region. Some QSEBs and UV brightenings, though  occurring close to each other, are not topologically connected with the  same reconnection process. The energy released during QSEBs falls in the  range 1023–1024 ergs.
Conclusions. This study shows that magnetic connectivity and  topological features, such as 3D null points, are crucial in linking  QSEBs in the lower atmosphere with UV brightenings in the transition  region.
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